Find the Derivative - d/dx arctan(3x)

Problem

d(arctan(3*x))/d(x)

Solution

1. Identify the outer function as arctan(u) and the inner function as u=3*x

2. Apply the chain rule, which states that (d(ƒ)*(g(x)))/d(x)=ƒ′*(g(x))⋅g(x)′

3. Recall the derivative of the inverse tangent function: d(arctan(u))/d(u)=1/(1+u2)

4. Differentiate the inner function u=3*x with respect to x which gives d(u)/d(x)=3

5. Substitute these components into the chain rule formula: 1/(1+(3*x)2)⋅3

6. Simplify the expression by squaring the term in the denominator and multiplying by the constant.

Final Answer

d(arctan(3*x))/d(x)=3/(1+9*x2)

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